Question

What is the missing number in the sequence $2,5,9,14,20,27,...$?
A. 37
B. 38
C. 39
D. 35
E. 41

Answer 1

Hint: We first denote the terms of the given sequence and try to find the sequence of the differences as ${{d}_{n}}={{t}_{n+1}}-{{t}_{n}}$. The sequence will be A.P. sequence. We try to find the next term of the new sequence and add that to the last term to find the missing solution.

Complete step-by-step solution:
We first try to find the differences between the sequences. We have $2,5,9,14,20,27,...$
We denote the differences as ${{d}_{n}}={{t}_{n+1}}-{{t}_{n}}$ where ${{t}_{n}}$ denote the ${{n}^{th}}$ term of the sequence.
We have \[{{t}_{1}}=2,{{t}_{2}}=5,{{t}_{3}}=9,{{t}_{4}}=14,...\].
Therefore,
\[\begin{align}
  & {{d}_{1}}=5-2=3 \\
 & {{d}_{2}}=9-5=4 \\
 & {{d}_{3}}=14-9=5 \\
 & {{d}_{4}}=20-14=6 \\
 & {{d}_{5}}=27-20=7 \\
\end{align}\]
We can see the sequence of the differences form an A.P. as $3,4,5,6,7,.....$
From the sequence we can say that the next term of the sequence $3,4,5,6,7,.....$ will be 8.
Therefore, the next term of the sequence $2,5,9,14,20,27,...$ will be $27+8=35$.
The correct option is D.

Note: We need to remember that we are not trying to find the sum as in that case we could have solved it by substituting the sequence with itself and finding the ${{n}^{th}}$ term. In that case the sequence gets shifted one time for the subtraction.